A035469 Triangle read by rows, the Bell transform of the triple factorial numbers A007559(n+1) without column 0.
1, 4, 1, 28, 12, 1, 280, 160, 24, 1, 3640, 2520, 520, 40, 1, 58240, 46480, 11880, 1280, 60, 1, 1106560, 987840, 295960, 40040, 2660, 84, 1, 24344320, 23826880, 8090880, 1296960, 109200, 4928, 112, 1, 608608000, 643843200
Offset: 1
Examples
Triangle starts:
{1}
{4, 1}
{28, 12, 1}
{280, 160, 24, 1}
{3640, 2520, 520, 40, 1}
References
- F. Bergeron, Ph. Flajolet and B. Salvy, Varieties of Increasing Trees, in Lecture Notes in Computer Science vol. 581, ed. J.-C. Raoult, Springer 1922, pp. 24-48.
Links
- Peter Bala, Generalized Dobinski formulas
- P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem, arXiv:quant-ph/0402027, 2004.
- Richell O. Celeste, Roberto B. Corcino and Ken Joffaniel M. Gonzales. Two Approaches to Normal Order Coefficients, Journal of Integer Sequences, Vol. 20 (2017), Article 17.3.5.
- Tom Copeland, A Class of Differential Operators and the Stirling Numbers
- Tom Copeland, Mathemagical Forests
- Tom Copeland, Addendum to Mathemagical Forests
- Milan Janjic, Some classes of numbers and derivatives, JIS 12 (2009) 09.8.3.
- Wolfdieter Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
- Wolfdieter Lang, First 10 rows.
- Shi-Mei Ma, Some combinatorial sequences associated with context-free grammars, arXiv:1208.3104 [math.CO], 2012. - From _N. J. A. Sloane_, Aug 21 2012
- E. Neuwirth, Recursively defined combinatorial functions: Extending Galton's board, Discrete Math. 239 (2001) 33-51.
- Mathias Pétréolle and Alan D. Sokal, Lattice paths and branched continued fractions. II. Multivariate Lah polynomials and Lah symmetric functions, arXiv:1907.02645 [math.CO], 2019.
Crossrefs
Programs
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Mathematica
a[n_, m_] /; n >= m >= 1 := a[n, m] = (3(n-1) + m)*a[n-1, m] + a[n-1, m-1]; a[n_, m_] /; n < m = 0; a[, 0] = 0; a[1, 1] = 1; Flatten[Table[a[n, m], {n, 1, 9}, {m, 1, n}]] (* _Jean-François Alcover, Jul 22 2011 *) rows = 9; a[n_, m_] := BellY[n, m, Table[Product[3k+1, {k, 0, j}], {j, 0, rows}]]; Table[a[n, m], {n, 1, rows}, {m, 1, n}] // Flatten (* Jean-François Alcover, Jun 22 2018 *)
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Sage
# uses[bell_matrix from A264428] # Adds a column 1,0,0,0, ... at the left side of the triangle. bell_matrix(lambda n: A007559(n+1) , 9) # Peter Luschny, Jan 19 2016
Formula
a(n, m) = Sum_{j=m..n} |A051141(n, j)|*S2(j, m) (matrix product), with S2(j, m):=A008277(j, m) (Stirling2 triangle). Priv. comm. to Wolfdieter Lang by E. Neuwirth, Feb 15 2001; see also the 2001 Neuwirth reference. See the general comment on products of Jabotinsky matrices given under A035342.
a(n, m) = n!*A035529(n, m)/(m!*3^(n-m)); a(n+1, m) = (3*n+m)*a(n, m) + a(n, m-1), n >= m >= 1; a(n, m) := 0, n < m; a(n, 0) := 0, a(1, 1)=1;
E.g.f. of m-th column: ((-1+(1-3*x)^(-1/3))^m)/m!.
From Peter Bala, Nov 25 2011: (Start)
E.g.f.: G(x,t) = exp(t*A(x)) = 1 + t*x + (4*t+t^2)*x^2/2! + (28*t + 12*t^2 + t^3)*x^3/3! + ..., where A(x) = -1 + (1-3*x)^(-1/3) satisfies the autonomous differential equation A'(x) = (1+A(x))^4.
The generating function G(x,t) satisfies the partial differential equation t*(dG/dt+G) = (1-3*x)*dG/dx, from which follows the recurrence given above.
The row polynomials are given by D^n(exp(x*t)) evaluated at x = 0, where D is the operator (1+x)^4*d/dx. Cf. A008277 (D = (1+x)*d/dx), A105278 (D = (1+x)^2*d/dx), A035342 (D = (1+x)^3*d/dx) and A049029 (D = (1+x)^5*d/dx).
(End)
Dobinski-type formula for the row polynomials: R(n,x) = exp(-x)*Sum_{k>=0} k*(k+3)*(k+6)*...*(k+3*(n-1))*x^k/k!. - Peter Bala, Jun 23 2014
Extensions
New name from Peter Luschny, Jan 19 2016
Comments